Congruent Triangles Explained

Two triangles are congruent when one is a rigid copy of the other: every corresponding side matches in length, and every corresponding angle matches in measure. Writing $△ABC\cong △DEF$ names that copy, with $A$ paired to $D$, $B$ to $E$, and $C$ to $F$.

You rarely measure all six parts. A short list of tests is enough: freeze three of them in the right pattern, and the rest are forced. That is how you prove two triangles are congruent. The live board below shows one of those patterns — SAS — so you can see the three pieces a proof must name.

What congruence names

Corresponding parts are the pairs that the congruence statement lines up. In $△ABC\cong △DEF$, side $AB$ corresponds to side $DE$, side $BC$ to side $EF$, side $CA$ to side $FD$, and the angles at $A,B,C$ correspond to the angles at $D,E,F$ in that order.

Congruent triangles have the same shape and the same size. Similar triangles share angles (and scaled sides) but need not share side lengths. This page stays with congruence — same size, not only the same angles.

The tests that work

Each test freezes three parts. The order in the name is the order around the figure.

1. SSS — three sides. If $AB=DE$, $BC=EF$, and $CA=FD$, the triangles are congruent.
2. SAS — two sides and the included angle between them. If $AB=DE$, $\angle B=\angle E$, and $BC=EF$, the triangles are congruent. The angle must sit between the two sides you named.
3. ASA — two angles and the included side between them.
4. AAS — two angles and a side opposite one of those angles. Two angles already force the third, so AAS is enough once a side is matched.
5. HL — hypotenuse and one leg, in a right triangle only. The right angles are the extra freeze that makes hypotenuse-leg work.

SSA is not a congruence test. Two sides and an angle opposite one of them can fit an acute triangle and an obtuse triangle with the same listed pieces. Classroom problems that look like SSA need a different route — often a right-angle freeze that turns the figure into HL, or an extra side that upgrades the data to SSS or SAS.

Worked example (SAS)

On the board, read the lower triangle as $△ABC$ and the upper as $△DEF$. The upper triangle is the lower one translated straight up, so the copy is forced by the construction, not drawn as a second guess.

• Side $AB$ matches side $DE$ (both bases have length $4$).
• The included angle at $B$ matches the included angle at $E$ (the marks $\alpha$ and $\beta$).
• Side $BC$ matches side $EF$.

That is the SAS package. With those three equalities named, the congruence $△ABC\cong △DEF$ is forced: $C$ corresponds to $F$, and the remaining sides and angles must match. A proof writes exactly those three equalities, in an order that makes the included angle obvious, then cites SAS.

If a problem instead gives all three sides, cite SSS. If it gives a right angle plus the hypotenuse and a leg, cite HL. The handbook page on triangles: area and perimeter is where the same three-sided figure is measured for area; this post is where you prove two of those figures are copies.

Drag the board, keep the test

$C$ is free on the lower triangle. Drag it. The upper triangle follows: $F$ is $C$ translated by the same vector, so it cannot stay behind. Side $AB$ stays equal to $DE$, the included angle at $B$ stays equal to the angle at $E$, and side $BC$ stays equal to $EF$. A proof does not need every length on the page. It needs those three SAS pieces, with the angle between the sides. The rest of each triangle is the rigid copy.

Degrees on classroom worksheets are the everyday unit. When a later formula needs $\pi$, the handbook page on degrees and radians on a live board builds that unit. Congruence itself does not care which unit you used, as long as corresponding angles were measured the same way.

GoodMaith keeps a relation like this as an object on a live board: name the three SAS pieces, drag a vertex, and see what a congruence statement is actually freezing. The answer was never only the letters SSS or SAS; it was seeing why those three pieces leave no room for a second, different triangle.